Plane sections of convex bodies of maximal volume

Abstract

Let K = {K0, . . . , Kk} be a family of convex bodies in Rn, 1 ≤ k ≤ n - 1. We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k-dimensional plane Ak ⊆ Rn, called a common maximal k-transversal of K, such that, for each i ∈ {0, . . . , k} and each x ∈ Rn, Vk(Ki ∩ Ak) ≥ Vk(Ki ∩ (Ak + x)), where Vk is the k-dimensional Lebesgue measure in Ak and Ak + x. Given a family K = {Ki}li=0 of convex bodies in Rn, l < k, the set Ck(K) of all common maximal k-transversals of K is not only nonempty but has to be "large" both from the measure theoretic and the topological point of view. It is shown that Ck(K) cannot be included in a v-dimensional C1 submanifold (or more generally in an (Hv, v)-rectifiable, Hv-measurable subset) of the affine Grassmannian AGrn,k of all affine k-dimensional planes of Rn, of O(n + 1)-invariant v-dimensional (Hausdorff) measure less than some positive constant Cn,k,l, where v = (k - l)(n - k). As usual, the "affine" Grassmannian AGrn,k is viewed as a subspace of the Grassmannian Grn+1,k+1 of all linear (k + 1)-dimensional subspaces of Rn+1. On the topological side we show that there exists a nonzero cohomology class θ ∈ Hn-k(Gn+1,k+1; Z2) such that the class θl+1 is concentrated in an arbitrarily small neighborhood of Ck(K). As an immediate consequence we deduce that the Lyusternik-Shnirel'man category of the space Ck(K) relative to Grn+1,k+1 is ≥ k - l. Finally, we show that there exists a link between these two results by showing that a cohomologically "big" subspace of Grn+1,k+1 has to be large also in a measure theoretic sense.